Theorem rp-7frege

Description: Distribute antecedent and add another. (Contributed by RP, 24-Dec-2019)

	Ref	Expression
Assertion	rp-7frege	$$⊢\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\theta \to \left(\left(\phi \to \psi \right)\to \left(\phi \to \chi \right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-frege2	$$⊢\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\left(\phi \to \psi \right)\to \left(\phi \to \chi \right)\right)$$
2		rp-frege24	$$⊢\left(\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\left(\phi \to \psi \right)\to \left(\phi \to \chi \right)\right)\right)\to \left(\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\theta \to \left(\left(\phi \to \psi \right)\to \left(\phi \to \chi \right)\right)\right)\right)$$
3	1 2	ax-mp	$$⊢\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\theta \to \left(\left(\phi \to \psi \right)\to \left(\phi \to \chi \right)\right)\right)$$